This study proves that the Diophantine equation (9d²+1)ˣ+(16d²-1)ʸ=(5d)ᶻ has a unique positive integer solution $(x,y,z)=(1,1,2)$, for all $d>1$. The proof employs elementary number theory techniques, including linear forms in two logarithms and Zsigmondy's Primitive Divisor Theorem, specifically when d is not divisible by $5$. In cases where d is divisible by $5$, an alternative method utilizing linear forms in p-adic logarithms is applied.
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Çokoksen et al. (2024) studied this question.
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